https://www.afr.com/technology/why-...od-in-the-origins-of-symmetry-20191118-p53bqy What this top mathematician finds in the origins of symmetry The winner of the 2019 Prime Minister's Prize for Science, Emeritus Professor Cheryl Praeger writes that she has often felt like an explorer discovering new realms. Professor Cheryl Praeger, winner of the 2019 Prime Minister's Prize for Science. Nov 23, 2019 — 12.15am Mathematics has always helped me to understand the world. At age six, entering my second year at Humpybong State School north of Brisbane, I was enormously relieved to learn about negative numbers. It had seemed completely illogical to me that I could not, for example, subtract 5 from 3, and now at last it was “allowed” and the answer minus 2 was acceptable! Throughout my schooling I loved learning more about maths and science. I grew up in various towns in the south-east of Queensland, as my dad worked for a bank and my family moved every few years. I was the oldest of three children and the only daughter, and the first in my extended family to go to university. I started at the University of Queensland in 1966 and chose a special honours stream in both maths and physics. It was a tough course but I loved it. From my second year I was the only girl in any of my classes. Symmetry can be found throughout nature, and is often taken as a sign of health. Spending a summer at the research mathematics department of the Australian National University in Canberra convinced me to aim for further maths studies. I won a scholarship to St Anne’s College at Oxford University in 1970 at the end of my undergraduate degree, and there I was introduced to the area that has captivated me for my entire career: the mathematics of symmetry. Symmetry is all around us. In the natural world and the built environment, from the spiral galaxies in the heavens to the same symmetry we see in spiral shells on the beach. In nature, symmetry is often taken as a sign of health: indeed the symmetry possessed by tiny organisms such as viruses allows them to reproduce successfully using only a very small amount of genetic material. It has been termed their genetic economy. Roman cauliflower is a famous example of a mathematical fractal structure in nature. In mathematics we measure symmetry using groups. Groups (of symmetries) distinguish otherwise similar objects. For example, a triangle with three equal sides has six symmetries. The centre is the point at equal distance from the three corners of the triangle, and rotating the triangle about its centre either one-third, two-thirds, or all the way around maps the triangle onto itself. These three symmetries are called rotations. Each of the other three symmetries comes from choosing one of the three corners and swapping the two sides next to it. It’s called a reflection, because it’s just like reflecting the triangle in a mirror placed on the line through the centre and the chosen corner with half of the triangle on either side. These six symmetries – three rotations and three reflections – form the group of the triangle. Similarly, a square has a symmetry group of size eight, having four rotations and four reflections. Each similar figure having all its sides of equal length has its own group, different from the group of every other figure in the family, so the groups for this family of “regular polygons” uniquely identify the individual members. They play a similar role to genetic DNA barcodes for distinguishing between plant species. Early in my career, in 1972, Danny Gorenstein began to harness the efforts of dozens of mathematicians worldwide to prove a hugely powerful mega-theorem. The theorem identifies all the building blocks or “atoms” from which every finite group can be constructed. They are called the finite simple groups, and while some of them are relatively easy to describe (like the ones involved in the group of the triangle) others are tremendously complicated. I was among the first mathematicians to exploit this watershed result to build new fundamental theory and new methods to study groups and symmetrical structures, such as networks used for communication, or designs used for scientific experiments. The symmetry groups I work with are used to model all kinds of natural phenomena. For example, the largest of the sporadic simple groups is called the Monster. It has approximately 8 x 10⁵³ elements and it has been speculated that it might be associated with quantum gravity and may even be the symmetry group of a black hole! Telescope image of Andromeda: symmetry can also be found the spiral galaxies in the heavens. New algorithms that I developed with my collaborators, Peter Neumann from Oxford University and Alice Niemeyer from RWTH Aachen University, are built into two major computer algebra systems – GAP and MAGMA. They run incredibly fast because they make deductions from a small amount of data and rely on the finite simple group classification for their correctness. The algorithms are designed to solve problems in many branches of mathematics involving symmetry. GAP and MAGMA are indispensable research tools used by mathematicians and scientists worldwide, as more than 10,000 published research articles testify. Often solving a mathematical problem requires techniques from outside my skill set and I will work collaboratively with several colleagues. Joint research like this can be really fun, learning from each other, as well as sharing the enthusiasm and jubilation of the final victory when we reach a solution. One exciting example of this is reported in a paper published last year in the Journal of the London Mathematical Society. A research team including Stephen Glasby and my graduate student, Kyle Rosa from the University of Western Australia, as well as Gabriel Verret from the University of Auckland, was searching for the best possible upper bound for a complexity measure of a finite symmetry group called “composition length”. Sometimes I can be so immersed in a really difficult mathematical problem that it takes over my life. Our results identified all the groups that actually attain our new upper bound. The fact that such groups exist at all told us that our bound was the best possible, and that we could find all of them was especially exhilarating. I felt like an explorer who has just made a fantastic discovery. Sometimes I can be so immersed in a really difficult mathematical problem that it takes over my life. I had tried many different attack strategies on one such problem about primitive permutation groups. The stress of working on this problem was so great that I needed to set some boundaries, and finally I decided I would give myself just one more year to find a solution: if unsolved by then, I would drop it. After some months my brain must have become so hard-wired into the intricacies of the problem that it continued to function through my subconscious while I was asleep. On several nights I awoke multiple times, and each time my mind was coursing through the logical arguments needed for the proof. The only way I could relax enough to get back to sleep was to feverishly make notes of these steps. Sometimes in the morning the reconstructed argument fell apart and I realised that my subconscious had ignored a vital mathematical complication. But eventually, one night, after several hours developing all the steps sketched in my notes, I obtained a complete proof. I was full of awe at what had just happened. The proof was so beautiful, and I felt grateful to have found it. I felt like I was genuinely uncovering a small new part of mathematical reality. Is high-dimensional mathematical space part of a pre-existing mathematical reality or simply a human construct? More than 2000 years ago, the Greek philosopher Plato believed that we “only explore what is already there”, and ever since mathematicians and philosophers have debated the question of whether humans create mathematical concepts and objects or discover them. Since the mathematical structures we study are often models of the natural world, we may uncover previously unknown relationships in nature. On the other hand, we may be building symmetrical structures in high dimensions that we cannot envisage in nature. Then it is a philosophical question as to whether this high-dimensional mathematical space is part of a pre-existing mathematical reality or simply a human construct. I do not expect all mathematicians to feel as I do: that their research uncovers parts of the universe, and thereby part of God’s creation. There is no consensus about this, and I would not expect there to be. In particular, I would not expect all mathematicians of faith – and as a Christian I am one – to feel as I do: that their research uncovers parts of the mathematical universe, and thereby part of God’s creation. There are many major research projects I am involved with and plan to tackle in the next years, in collaboration with my colleagues across the globe, that will help us understand better the mathematics of symmetry – and in turn, perhaps, the natural symmetry of the universe. Cheryl Praeger is the recipient of the 2019 Prime Minister's Prize for Science.